Hi everyone! I've just passed a question ? 129 from diagnostic questions for Exam 1:
Imagine two perpetual bonds, ie bonds that pay a coupon till perpetuity and the issuer does not have an obligation to redeem. If the coupon on Bond A is 5%, and on Bond B is 15%, which of the following statements will be true:
I. The Macaulay duration of Bond A will be 3 times the Macaulay duration of Bond B.
II. Bond A and Bond B will have the same modified duration
III. Bond A will be priced at less than 1/3rd the price of Bond B
IV. Both Bond A and Bond B will have a duration of infinity as they never mature

The right answer is II. Could anyone explain me why it is so? Since MD=1/rc and rc- is a current yield of a bond which depends on a coupon payment (Coupon/Price), then modifying duration of Bond B is supposed to be 3 times modifying duration of Bond A...
Thanks in advance;)

Let us try an intuitive interpretation - forget the formulae for the moment.

Consider the bonds from the perspective of the investor. Because these are perpetual, an investor will never see a dime of capital back. All they will get is interest each year. In that sense, the bonds represent an annuity that continues forever. If the coupon is 5%, then it provides a cash flow of $5 each year. The bond with a 15% coupon provides a cash flow of $15 each year. For the investor, the second bond is worth exactly 3x the first one. He or she can get an equivalent of the second bond by buying three of the first. So the price of three bonds each with a coupon of 5% will be the same as the price of a single 15% bond. In the end, each bond will end up with the same yield as the price will move to equalize the realized yield on the investment.

Now the Macaulay duration is 1/yield. The 'yield' will be the same on the two bonds, because even if the 15% coupon bond provides 3 times the cash each year, it will be three times as expensive as well bringing its yield down to the same level as that for the bond with the 5% coupon. Thus the Macaulay duration for both bonds will be the same.